It J Pure Appl Sc Techol, () (00), pp 79-86 Iteratoal Joural of Pure ad Appled Scece ad Techology ISSN 9-607 Avalable ole at wwwjopaaat Reearch Paper Some Stroger Chaotc Feature of the Geeralzed Shft Map Idral Bhaum,* ad Baya S Choudhury, Departmet of Mathematc, Begal Egeerg ad Scece Uverty, Shbpur, Howrah - 703, Wet Begal, Ida * Correpodg author, e-mal: (raadra006@yahooco) (Receved: --00 ; Accepted: 7--00) Abtract: Recetly, we have troduced the oto of the geeralzato of the hft map, that, the geeralzed hft map, the ymbol pace Σ I th paper we have proved ome troger chaotc properte of the geeralzed hft map A example ha bee gve the lat ecto Keyword: Chao, Symbol pace, Geeralzed hft map, Topologcally mxg, Chaotc depedece o tal codto Itroducto A dyamcal ytem a tudy of how phycal ad mathematcal ytem evolve wth tme, developed through the collectve effort of mathematca ad cett may deferet feld A dyamcal ytem clude the followg compoet: a phae pace S whoe elemet repreet poble tate of the ytem; tme t (whch may be dcrete or cotuou) ad a evoluto law (that, a rule that allow determato of the tate at tme t from the owledge of the tate at all prevou tme) Hece a geeral dyamcal
It J Pure Appl Sc Techol, () (00), 79-86 80 ytem ca be defed a a par ( X, f ) cotg of a et X together wth a cotuou map f from X to telf Chaotcty a mportat property for ay dyamcal ytem The tudy of chaotc dyamc ha become creagly popular at the preet day Although there ha bee o uverally accepted mathematcal defto of chao, t geerally beleved that f for ay ytem the dtace betwee the earby pot creae ad the dtace betwee the far away pot decreae wth tme, the ytem ad to be chaotc Hece a dyamcal ytem chaotc f the orbt of t (or a ubet of t) are cofed to a bouded rego, but tll behave upredctably Th prt caught everal equvalet defto [6, 7, 8, 9, 0, ] of chao The term chao wa frt ued mathematcally by L ad Yor ther paper Perod three mple chao [9] 975 Symbolc dyamcal ytem ( Σ, σ ) ad ( Σ, σ ), where Σ the equece pace, σ the hft map ad σ the geeralzed hft map, are alo example of chaotc dyamcal ytem I partcular there are everal wor o ymbolc dyamc where dyamc are repreeted by map o ymbol pace Some of thee wor are oted referece [,, 3, 4, 7, 8,, ] Of partcular teret the pace Σ whch ha bee codered a large umber of wor, where = α : α = ( α α ), α 0 or }, a metrc pace { 0 = wth the metrc (, ) = t d t, where = ( + ) = 0 0 ad t = ( t t 0 ) are two pot of Σ It eay to prove that, by our choe metrc the maxmum dtace betwee ay two pot of Σ Recetly, the preet author troduced the cocept of geeralzed hft map [] I th paper we have proved ome troger chaotc properte of the geeralzed hft map I Theorem 3, t proved that the geeralzed hft map topologcally mxg o Σ The we have proved, Theorem 3, that the geeralzed hft map ha chaotc depedece o tal codto We alo have gve a example of a cotuou fucto whch topologcally tratve but ot chaotc the ee of Du [8] Mathematcal Prelmare Here we gve ome defto ad lemma whch are requred for ext two ecto
It J Pure Appl Sc Techol, () (00), 79-86 8 Defto (Shft map [7]) The hft map σ : Σ Σ defed by σ ( α) = ( α α ), where α = ( α 0 α) ay pot of Σ Defto (Geeralzed hft map []) The geeralzed hft map σ : Σ Σ defed by σ ( ) = ( + ), where = ( 0 ) ay pot of Σ ad ay teger Defto 3 (Topologcally tratve [7]) Let ( X, ρ) be a compact metrc pace A mappg f : X X ad to be topologcally tratve f for ay par of o -empty ope et K, L X there ext 0 uch that f ( K) L φ Defto 4 (Topologcally mxg []) Let ( X, ρ) be a compact metrc pace ad f : X X be a cotuou map The map f called topologcally mxg f for ay two o-empty ope et U, V X there ext m 0 uch that for all m, f ( U ) V φ Defto 5 (Setve depedece o tal codto [7]) Let ( X, ρ) be a compact metrc pace A cotuou map f : X X ha etve depedece o tal codto f there ext δ > 0 uch that, for ay x S ad ay eghborhood N (x) of x there ext y N(x) ad 0 uch that ρ ( f ( x), f ( y)) > δ Defto 6 (L -Yore par [5]) A par (, ) X x y called a L -Yore par (wth modulu δ ) f p p Lt Sup ρ( f ( x), f ( y)) δ p p p ad Lt If ρ ( f ( x), f ( y)) = 0, p where ( X, f ) a dyamcal ytem, X beg a compact metrc pace wth the metrc ρ ad f a cotuou mappg o X Defto 7 (Chaotc depedece o tal codto [5]) A dyamcal ytem ( X, f ) ha chaotc depedece o tal codto f for ay x X ad ay eghborhood N (x) of x there y N(x) uch that the par (, ) X x y L -Yore We alo eed the followg lemma
It J Pure Appl Sc Techol, () (00), 79-86 8 Lemma [7]: Let, t Σ ad = t, for = 0,,, m The d(, t) < m ad coverely f d(, t) < the m = t, for = 0,,, m 3 The Ma Theorem Theorem 3 The geeralzed hft map σ : Σ Σ topologcally mxg o Σ Proof: We tae ay two o -empty ope et U ad V of Σ Let be ay pot uch that m { ( u, β ) } = ε u = ( u u ) U d, for ay β belog to the boudary of the et U 0 ad v = ( v v ) V m β = ε d ( v, ), for ay 0 be ay pot uch that { } β belog to the boudary of the et V, where ε, ε > 0 We ow chooe two potve teger ad <ε uch that ad <ε Latly, we coder the equece of pot gve ( ) by, α u u u (0) v v v ), for =,3, = ( 0 0 ad α u u u v v v ) = ( 0 0 Now, d ( u, α ) < < ε, =,,, by Lemma (3) Hece α U, =,,, that, σ ( α ) σ ( U ), for ay 0 O the other had, σ ( α ) = ( v0v v ) Hece d ( σ ( α), v) < < ε, by applyg Lemma aga (3) Th gve σ ( α ) V
It J Pure Appl Sc Techol, () (00), 79-86 83 I vrtue of (3) ad (3) we ca ay that σ ( U ) V φ Next we coder the pot + α The σ α ) ( v v v ) Whch aga belog to V Hece ( = 0 + σ ( U ) V φ Cotug th proce by tag allα we get σ ( U V φ, for ) all Hece σ topologcally mxg o Σ Theorem 3 The dyamcal ytem ( Σ,σ ) ha chaotc depedece o tal codto Proof: At frt we gve ome otato whch help u to prove Theorem 3 Let = ( 0 ) be ay pot of Σ ad U be ay ope eghborhood of Let S = 0 ad P = p p 0 pm be two fte equece of 0 ad, the S P = p Further, f we uppoe that T, 0 p0 p, T p are p fte equece of 0 ad ; T T Tp maer a above 3 If β ay bary umeral, we deote the complemet of β = 0 or, the β = or 0 m, T ca be defed a mlar β by 4 Let F, 0) = ( ) ( + 3 3+ 4 4 4+ 5 F ( ) (, ) + = 5 5+ 6+ 6+ 6+ + 7+ β That, f,, ad o o Note that for ay eve teger m, F (, + m) a fte trg of legth ( + m) 5 Latly, we tae t Σ uch that ( (0) () F (, + 0) F (, + ) F (, 4) ) t = 0 +, where ( α ) = αα α tme We coder the pot ad the ope eghborhood U of defed the above otato Sce U ope we ca alway chooe a > 0 ε, uch that m{ d (, α )} = ε, for ay
It J Pure Appl Sc Techol, () (00), 79-86 84 α belog to the boudary of the et U We chooe o large that < ε By our cotructo ad t agree up to Hece d (, t) < < ε, by Lemma So 3 σ ad t) ( ) 3 t U Now ) ( ) ( = 3 3 + 4 σ ( = 3 3 + 4 Note that t cot of ftely may fte equece of the type A (, + m) So we get Lt Sup d( σ ( ), ( )) (( 3 4 ), ( 3 σ t Lt d 4 )) Lt ( + + + ) = (33) Hece, Lt Sup d( σ ( ), σ ( t)) = 4 Smlarly, ) ( ) Aga we get that ( = 4 4+ 5 4 σ ad σ t) ( ) ( = 4 4+ 5 (( ), ( )) Lt If d( σ ( ), σ ( t)) Lt 4 5 4 5 0 0 0 Lt ( + + + ) = 0 (34) Hece, Lt If d( σ ( ), σ ( t)) = 0 From (33) ad (34) t proved that the par (, t) L -Yore Hece the dyamcal ytem ( Σ,σ ) ha chaotc depedece o tal codto 4 Cocluo I th paper we have proved ome troger chaotc properte of the geeralzed hft map Alo the property Defto 6 very mportat for ay dyamcal ytem, becaue th property maly baed o L -Yore par but ha ome commo feature of etve depedece o tal codto Hece we ca ay that the geeralzed hft map ha a property whch baed o L -Yore par but have ome commo feature of etve depedece o tal codto Alo we have proved that the geeralzed hft
It J Pure Appl Sc Techol, () (00), 79-86 85 map topologcally mxg o Σ, whch a property troger tha topologcal tratvty Latly, we gve a example of a cotuou fucto whch topologcally tratve but ot chaotc the ee of Du [8] Example 4: Let f :[,] [,] be a fucto defed by f ( x) = 7 7 x +, x 6 6 7 7x, x 0 7 x, 0 x The fucto defed above obvouly a cotuou fucto It ca be ealy proved that the fucto topologcally tratve But t ot chaotc the ee of Du ce the perod two pot cloe to each other 7 ad the cloed terval [ 0,] are jumpg alteratvely ad ever get 3 Acowledgemet Idral Bhaum acowledge h father Mr Sadha Chadra Bhaum for h help preparg the maucrpt Referece [] I Bhaum ad B S Choudhury, Dyamc of the geeralzed hft map, Bull Cal Math Soc, 0(5) (009), 463-470 [] I Bhaum ad B S Choudhury, The hft map ad the ymbolc dyamc ad applcato of topologcal cojugacy, J Phy Sc, 3 (009), 49-60 [3] I Bhaum ad B S Choudhury, Topologcally cojugate map ad ω -chao ymbol pace, It J Appl Math, 3() (00), 309-3 [4] I Bhaum ad B S Choudhury, Some uow properte of ymbolc dyamc, It J Appl Math, 3(5) (00), 94-949
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